Subgroups — #30761636

Order 17152.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562616	C₂	1	2	yes	maximal
2	C₂×#16863153	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524002	C₂²	1	4	yes	—
5	#16863266	—	2	2	no	—
6	#16863262	—	2	2	no	—
7	#16863256	—	2	4	no	—
8	C₄×C₄⋊Dic₆₇	C₄	1	8	yes	complemented
9	#16863153	—	2	8	no	—
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₆₇	—	2	2	no	—
13	C₄×M₁₆×C₆₇	—	2	2	no	—
14	C₂×#97×C₆₇	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
16	#14524002	—	2	8	no	—
17	#14524002	—	2	8	no	—
18	C₂×C₄⋊Dic₆₇	—	2	8	no	—
19	C₂×C₄⋊Dic₆₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄×Dic₆₇	C₂×C₄	1	8	yes	—
22	C₆₇⋊#95	—	2	8	no	—
23	C₂³×C₄×C₆₇	D₄	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₆₇	—	2	4	no	—
26	C₂×M₁₆×C₆₇	—	2	4	no	—
27	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
28	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
36	#97×C₆₇	—	2	4	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₂²×Dic₆₇	—	2	8	no	—
40	C₄⋊Dic₆₇	—	4	16	no	—
41	C₄⋊Dic₆₇	—	4	8	no	—
42	C₂²⋊Dic₆₇	—	4	4	no	—
43	C₂²⋊Dic₆₇	—	4	4	no	—
44	C₄×Dic₆₇	—	2	16	no	—
45	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₆₇	#41	1	4	yes	—
58	M₁₆×C₆₇	—	2	8	no	—
59	M₁₆×C₆₇	—	2	8	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	complemented
63	C₄²×C₆₇	C₄²	1	2	yes	—
64	C₄²×C₆₇	C₄²	1	2	yes	—
65	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
70	C₂³×C₆₇	—	4	2	no	—
71	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₆₇	#124	1	2	yes	—
74	C₂³×C₆₇	#127	1	1	yes	characteristic
75	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	8	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	#95	1	2	yes	—
81	C₂×C₄×C₆₇	#95	1	4	yes	—
82	C₂×C₄×C₆₇	#95	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₇	#117	1	2	yes	—
88	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
90	C₂×C₄×C₆₇	#121	1	4	yes	—
91	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
92	C₂×C₄×C₆₇	#125	1	2	yes	—
93	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
94	C₂×Dic₆₇	—	4	16	no	—
95	C₂×Dic₆₇	—	4	8	no	—
96	C₈×C₆₇	—	2	8	no	—
97	C₂²×C₆₇	—	2	4	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	4	2	no	—
100	C₂²×C₆₇	—	4	1	no	—
101	C₂²×C₆₇	—	4	1	no	—
102	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
103	C₂²×C₆₇	#382	1	1	yes	characteristic
104	C₂²×C₆₇	#384	1	1	yes	characteristic
105	C₂²×C₆₇	#390	1	2	yes	—
106	C₂²×C₆₇	#391	1	2	yes	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	H(3,ℤ/4)	1	4	yes	complemented
110	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₆₇	#380	1	2	yes	—
112	C₄×C₆₇	#389	1	4	yes	—
113	Dic₆₇	—	8	8	no	—
114	#10837	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	—	2	2	no	—
116	C₂×C₆₇	—	2	1	no	—
117	C₂×C₆₇	—	2	1	no	—
118	C₂×C₆₇	—	2	2	no	—
119	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
120	C₂×C₆₇	C₂×H(3,ℤ/4)	1	2	yes	—
121	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
122	#2285	—	67	2	no	—
123	C₂×H(3,ℤ/4)	—	67	4	no	—
124	C₆₇	#10837	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
126	C₂×#117	—	67	4	no	—
127	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
128	C₄×M₁₆	—	2	2	no	—
129	C₄×M₁₆	—	2	2	no	—
130	C₂×#97	Dic₆₇	1	2	yes	complemented
131	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
132	#384	—	134	2	no	—
133	#382	—	134	2	no	—
134	#380	—	134	4	no	—
135	C₄×C₄⋊C₄	—	67	8	no	—
136	H(3,ℤ/4)	—	134	8	no	—
137	C₂³×C₄	D₂₆₈	1	2	yes	—
138	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
142	C₂×M₁₆	D₂₆₈	1	4	yes	—
143	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
144	#117	—	134	8	no	—
145	#117	—	134	8	no	—
146	C₂×C₄⋊C₄	—	134	8	no	—
147	C₂×C₄⋊C₄	—	134	4	no	—
148	C₂×C₂²⋊C₄	—	134	4	no	—
149	C₂×C₄²	—	67	8	no	—
150	C₂×C₄²	Dic₁₃₄	1	2	yes	—
151	C₂×C₄²	C₄×D₆₇	1	4	yes	—
152	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	134	8	no	—
160	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	134	8	no	—
163	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
164	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
165	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
166	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
168	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
170	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
171	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
173	C₂²×C₄	#11761628	1	4	yes	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	complemented
179	C₄⋊C₄	—	268	16	no	—
180	C₄⋊C₄	—	268	8	no	—
181	C₂²⋊C₄	—	268	4	no	—
182	C₂²⋊C₄	—	268	4	no	—
183	C₄²	—	134	16	no	—
184	C₄²	C₄×Dic₆₇	1	2	yes	—
185	C₄²	C₄×Dic₆₇	1	2	yes	—
186	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
189	C₄²	Dic₆₇⋊C₄	1	4	yes	—
190	C₄²	Dic₆₇⋊C₄	1	4	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
193	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
194	C₂³	#14524031	1	2	yes	—
195	C₂³	#14524040	1	1	yes	characteristic
196	C₂³	#14524043	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	268	16	no	—
202	C₂×C₄	—	268	8	no	—
203	C₂×C₄	C₆₇⋊#95	1	2	yes	—
204	C₂×C₄	C₆₇⋊#95	1	4	yes	—
205	C₂×C₄	C₆₇⋊#95	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
210	C₂×C₄	#14524002	1	2	yes	—
211	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
212	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
213	C₂×C₄	#14524017	1	4	yes	—
214	C₂×C₄	#14524022	1	4	yes	—
215	C₂×C₄	#14524035	1	2	yes	—
216	C₂×C₄	#14524043	1	2	yes	—
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
224	C₂²	#16863262	1	1	yes	characteristic
225	C₂²	#16863266	1	1	yes	characteristic
226	C₂²	#16863298	1	2	yes	—
227	C₂²	#16863302	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	536	8	no	—
231	C₄	#16863153	1	4	yes	complemented
232	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
233	C₄	#16863256	1	2	yes	—
234	C₄	#16863294	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	#22558439	1	1	yes	characteristic
240	C₂	C₂×#16863153	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#10837
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. C₄×C₄⋊Dic₆₇	C₄	semidirect
14. C₂×#97×C₆₇	C₄	semidirect
15. C₂×C₄×C₈×C₆₇	C₄	semidirect
62. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
109. C₄×C₆₇	H(3,ℤ/4)	semidirect
124. C₆₇	#10837	semidirect
130. C₂×#97	Dic₆₇	semidirect
131. C₂×C₄×C₈	Dic₆₇	semidirect
178. C₂×C₈	C₄⋊Dic₆₇	semidirect
231. C₄	#16863153	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562616	2
C₂×#16863153	2
C₂×C₄×M₁₆×C₆₇	2
#10837	67